Talk by Matías Martres: Even Unimodular Lattices through their Theta Functions - May 22, 2026

The talk will be held as part of the IMERL Algebra Seminar, presented by Lic. Matías Martres.

Date, Time and Venue: Friday, May 22, 2026, in the Seminar Room at IMERL, Facultad de Ingeniería.

Ttitle: Even Unimodular Lattices through their Theta Functions

Abstract: Lattices are intimately related to quadratic forms. Within this category, even unimodular lattices are of particular interest; for instance, they are the underlying structures of the optimal sphere packings by unit balls in dimension 8 (Viazovska, 2016) and dimension 24 (Viazovska et al., 2016).

After reviewing the basic definitions, we will present the central object of this talk: the theta series associated with a lattice. Constructed from combinatorial information —norms of lattice elements— this series enables us to define a function on the upper-half plane with nice analytic properties. This allows us to show that, for example, the existence of an even unimodular lattice imposes restrictions on the ambient space.

Finally, we will tackle the classification problem. To this end, we will see how the Smith-Minkowski-Siegel mass formula can be used to solve the problem in low dimension, as well as to understand that an exhaustive classification in dimension greater than or equal to 32 is a simply unmanageable task. We will close by outlining the classification in dimension 24 due to Niemeier (1973), following Venkov’s simplification of the proof (1978) where root systems, theta functions and algebraic codes come together.